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DifferentialIntegralMultivariable

Common difficulties

Visualizing surfaces and regions
Selecting coordinates and bounds
Knowing when a theorem or method applies

Hold one coordinate fixed

This original example practises partial derivatives. Use your section’s outline to decide which topics to review.

On a small plate, let $x$ and $y$ be numerical coordinates measured in metres, with $0\le x\le3$ and $0\le y\le3$ . The numerical temperature in degrees Celsius is $T(x,y)=18+x^2+3xy-2y^2$ . At $P=(1,2)$ , find $T_x(P)$ , the local temperature change per metre as $x$ increases while $y$ stays fixed, and $T_y(P)$ , the corresponding rate as $y$ increases while $x$ stays fixed. Predict whether the temperature initially increases or decreases along each move. Check the first rate using a one-variable slice.

Show a hint

For the x derivative, treat every occurrence of y as a constant. To check it, substitute y=2 before differentiating the resulting one-variable function.

Show the reasoning

1. Keep the fixed coordinate in the formula. Holding $y$ fixed gives $T_x=2x+3y$ : the term $-2y^2$ is constant with respect to $x$ . Holding $x$ fixed instead gives $T_y=3x-4y$ . The subscripts name the coordinate being varied.

2. Evaluate at the point. Thus $T_x(1,2)=8$ and $T_y(1,2)=-5$ , both in degrees Celsius per metre. Increasing $x$ with $y=2$ initially raises temperature; increasing $y$ with $x=1$ initially lowers it. These are local rates, not temperature readings.

3. Check the slice. Along $y=2$ , define $g(x)=T(x,2)=10+x^2+6x$ . A prime denotes a one-variable derivative, so $g^\prime(1)=2(1)+6=8$ . Also $T(1,2)=17$ . For a nonzero change $h$ in the numerical x-coordinate, with $1+h$ still in the plate, expansion gives

$\frac{T(1+h,2)-T(1,2)}{h}=8+h.$

Letting $h$ approach zero recovers 8. Changing both coordinates would describe another path; it would not calculate this partial derivative.

Your turn: change the point and coordinate

Use the same plate and temperature model, but start at $Q=(2,1)$ . Hold $x=2$ and move to $y=0.9$ . Predict whether the temperature initially rises or falls. Find $T_y(Q)$ , use it to estimate the temperature change, then calculate the exact change. Here $\Delta y=0.9-1=-0.1$ is the signed coordinate change in metres. Explain why the estimate and exact result differ.

Check your reasoning

At $Q$ , $T_y=3(2)-4(1)=2\,{}^\circ\mathrm{C/m}$ . Since the movement has negative $\Delta y$ , the local estimate is $2(-0.1)=-0.20\,{}^\circ\mathrm C$ : the temperature decreases along this move. The positive partial derivative describes increasing $y$ ; it does not say that temperature increases along every movement.

On the slice $x=2$ , $T(2,y)=22+6y-2y^2$ . Thus $T(2,1)=26$ and $T(2,0.9)=25.78$ , giving an exact change of $-0.22\,{}^\circ\mathrm C$ .

For a change $h$ from $y=1$ , expansion gives $T(2,1+h)-T(2,1)=2h-2h^2$ . The derivative estimate keeps the linear term $2h$ . At $h=-0.1$ , the omitted term is $-0.02\,{}^\circ\mathrm C$ , explaining the difference. Both points stay inside the model’s domain.

A tutor can ask which coordinate is fixed, compare your partial derivative with the one-variable slice, and help you keep a local rate separate from a finite temperature change. Then try the changed-point question without the solution.

Practise regions for double integrals For a separate multiple-integral chapter, read the slice guide and choose a region question that matches your outline.
Dawson’s mathematics course descriptions 201-SNC-DW lists partial derivatives and multiple integrals. A catalogue description does not confirm a particular term or section.

How tutoring can help

We confirm support and availability for your chapter before booking. A tutor can compare a partial-derivative calculation with its one-variable check, help identify any mistake in your calculation and distinguish a local temperature rate from a temperature change. Together, you can use the example above or a question you choose.

Exam preparation for 201-SNC-DW

If partial derivatives are on your review list, retry the changed-point question without its solution. State which coordinate stays fixed, predict the sign of the temperature change, then compare the derivative estimate with the exact change. Explain the difference and check the units of both results.

Use your section’s outline to choose the chapters to review. We work through permitted practice and comparable problems, without completing graded assignments or exams on your behalf.

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